Unified Decision Procedures for Regular Expression Equivalence

Transcription

1 Unified Decision Procedures for Regulr Expression Equivlence Tois Nipkow nd Dmitriy Trytel Fkultät für Informtik, Technische Universität München, Germny Astrct. We formlize unified frmework for verified decision procedures for regulr expression equivlence. Five recently pulished formliztions of such decision procedures (three sed on derivtives, two on mrked regulr expressions) cn e otined s instnces of the frmework. We discover tht the two pproches sed on mrked regulr expressions, which were previously thought to e the sme, re different, nd we prove quotient reltion etween the utomt produced y them. The common frmework mkes it possile to compre the performnce of the different decision procedures in meningful wy. 1 Introduction Equivlence of regulr expressions is perennil topic in computer science. Recently it hs spwned numer of formlized nd verified decision procedures for this tsk in interctive theorem provers [3, 6, 10, 19, 21]. Except for the formliztion y Brint nd Pous [6], ll these decision procedures operte directly on vritions of regulr expressions. Although they (implicitly) uild utomt, the sttes of the utomt re leled with regulr expressions, nd there is no glol trnsition tle ut the nextstte function is computle from the regulr expressions. The motivtion for working with regulr expressions is simplicity: regulr expressions re free dttype which proof ssistnts nd their users love ecuse it mens induction, recursion nd equtionl resoning the core competence of proof ssistnts nd functionl progrmming lnguges. Yet ll these decision procedures sed on regulr expressions look very different. Of course, the next-stte functions ll differ, ut so do the ctul decision procedures nd their correctness, completeness nd termintion proofs. The contriutions of our pper re the following: A unified frmework (Sect. 3) tht we instntite with ll the ove pproches (Sects. 4 nd 5). The frmework is simple reflexive trnsitive closure computtion tht enumertes the sttes of product utomton. Proofs of correctness, completeness nd termintion tht re performed once nd for ll for the frmework sed on few properties of the next-stte function. A new perspective on prtil derivtives tht recsts them s Brzozowski derivtives followed y some rewriting (Sect. 4). The discovery tht Asperti s lgorithm is not the one y McNughton-Ymd [20], s stted y Asperti [3], ut dul construction which pprently hd not een considered in the literture nd which produces smller utomt (Sect. 5). An empiricl comprison of the performnce of the different pproches (Sect. 6). The discussion of relted work is distriuted over the relevnt sections of the pper.

2 2 Preliminries Iselle/HOL is sed on Church s simple type theory (see [22, Prt I] for recent introduction). Types τ re uilt from type vriles α, β, etc. vi function types, other type constructors re written postfix. The nottion t :: τ mens tht term t hs type τ. Types αset nd αlist re the types of sets nd lists of elements of type α. They come with the following voculry: function set (conversion from lists to sets), [] (empty list), # (list (ppend), hd (hed), tl (til) nd mp. Recursive functions over dttypes re executle, nd Iselle cn generte from them code in functionl lnguges [15]. This includes functions on finite sets [14]. Unless stted otherwise ll functions in this pper re executle. Locles [4] re Iselle s tool for modelling prmeterized systems. A locle fixes prmeters nd sttes ssumptions out them: locle A = fixes x 1 nd... nd x n ssumes n 1 : P 1 x nd... nd n m : P m x In the context of the locle A, we cn define constnts tht depend on the prmeters x i nd prove properties out those constnts using the ssumptions P i (ccessed under the nme n i ). Prmeters cn e instntited: interprettion J: A where x 1 = t 1... x n = t n. The commnd issues proof oligtions P i t (tht the user must dischrge) nd exports constnts nd theorems from the locle with x i instntited to t i. Multiple interprettions of the sme locle re possile; the prefix J. dismigutes different instnces. Regulr expressions re defined s recursive dttype: dttype αrexp = 0 1 A α αrexp + αrexp αrexp αrexp (αrexp) with the usul (non-executle) semntics L :: α rexp α lng, where α lng is short for (α list) set. In concrete regulr expressions, we sometimes omit the constructor A for redility. The recursive function nullle :: α rexp ool stisfies nullle r [] L r. The functions Σ :: αrexp αset nd toms :: αrexp αlist compute the set nd list of toms (the rguments of constructor A) in regulr expression. The (non-executle) left quotient of lnguge L :: α lng w.r.t. some :: α is defined y D L = {w # w L}. The extension of D from single symols to words w :: αlist cn e expressed s fold D w where fold :: (α β β) αlist β β. 3 Regulr Expression Equivlence Frmework Regulr expression (lnguge) equivlence is usully reduced to (lnguge) equivlence of utomt. In principle our frmework does the sme, except tht we construct the utomt on the fly nd replce the trditionl trnsition tle y computtions on regulr-expression-like ojects. We strt y relting regulr expressions nd utomt. Left quotients of regulr lnguge L cn e understood s sttes of deterministic utomton M L with the initil stte L, the trnsition function D, nd the ccepting sttes eing those lnguges K for which [] K holds. This utomton (restricted to rechle sttes) is finite nd miniml y the Myhill-Nerode theorem. 1 The following locle cptures this left-quotient-sed view of n utomton: 1 Note tht the Myhill-Nerode reltion L cn e defined s v L w fold D v L = fold D w L. The quotient of M L y this reltion is isomorphic to M L ; hence M L is miniml.

3 locle rexpda = fixes ι :: αrexp σ nd L :: σ αlng nd δ :: α σ σ nd o :: σ ool ssumes ιl: L(ι r) = L r nd δl: L(δ s) = D (L s) nd ol: o s [] L s The prmeters ι nd L formlize wht regulr-expression-like mens: ι r emeds the regulr expression r into stte of type σ, wheres L gives elements of σ lnguge semntics, which coincides with the lnguge semntics of regulr expressions y the ssumption ιl. The function δ is the symolic computtion of left quotients on σ ccording to δl. It cn e regrded s the trnsition function of n utomton with sttes in σ nd the initil stte ι r. Accepting sttes of this utomton re given y o. Let us develop nd verify some lgorithms in the context of rexpda. For strt, regulr expression mtching is esy to define mtch r w = o (fold δ w (ι r)) nd prove correct: mtch r w w L r. Now we tckle the equivlence checker. We follow the well-known product utomton construction where lnguge equivlence mens o s 1 o s 2 for ll sttes (s 1, s 2 ) of the product utomton. Alterntively, one cn view this procedure s the construction of isimultion reltion etween two utomt: lnguge equivlence nd existence of isimultion coincide for deterministic utomt [24]. The set of rechle sttes of n utomton cn e otined s the reflexive trnsitive closure of the strt stte under λ p. mp (λ. δ p) s where s :: αlist is the lphet. We define reflexive trnsitive closure opertion rtc :: (α ool) (α αlist) α (αlist αset)option where type αoption is the dttype None Some α. It is used to encode whether the closure is finite (Some is returned) or infinite (None is returned). The function rtc is defined using while comintor nd is executle (provided its rguments eing executle); the result Some corresponds to terminting computtion [19]. The definition cn e found in Iselle/HOL s lirry theory While_ Comintor under its full nme rtrncl_ while. The prmeters nd result of rtc p next strt hve the following mening: Predicte p is test tht stops the closure computtion if n element not stisfying p is found; this is merely n optimiztion. Function next mps n element to list of successors. Of course strt is the strt element. A result Some (ws, Z) mens tht the closure computtion terminted with worklist ws nd set of rechle elements Z. If ws is empty, Z is the set of ll elements rechle from strt; otherwise, the computtion ws stopped ecuse n element not stisfying p ws found. More precisely, we proved rtc p next strt = Some (ws, Z) = if ws = [] then Z = R ( z Z. p z) else p (hd ws) hd ws R (1) where R = {(x, y) y set (next x)} {strt} nd is infix reltion ppliction: r {x} = {y (x, y) r}. The stte spce of the product utomton is computed s follows: closure :: αlist σ σ ((σ σ)list (σ σ)set)option closure s = rtc (λ(s, t). o s ot) (λ(s, t). mp (λ. (δ s, δ t)) s)

4 The predicte λ(s, t). o s ot stops the computtion s soon s contrdiction to lnguge equlity is found. The ctul lnguge equivlence checker merely needs to test if the worklist is empty t the end: eqv :: αrexp αrexp ool eqv r s = cse closure (toms toms s) (ι r, ι s) of Some ([], _) True _ Flse The lphet given to closure is the conctention of the toms in the two expressions. Soundness of eqv is n esy consequence of the following property, which in turn follows from (1): closure (toms toms s) (ι r,ι s) = Some (ws, Z) = ws = [] L r = L s Theorem 1 (in rexpda). eqv r s = L r = L s. This is prtil correctness sttement ecuse it ssumes tht the cll to closure in eqv returns Some, i.e. termintes. Termintion of closure needs finiteness of the underlying utomton. Therefore we extend rexpda with n explicit ssumption of finiteness: locle rexpdfa = rexpda + ssumes fin: finite {fold δ w (ι r) w :: αlist} In this context the termintion lemm for closure is n esy consequence of fin nd the following termintion property of rtc: finite ({(x, y) y set ( f x)} {x}) = y. rtc p f x = Some y Lemm 2 (in rexpdfa). closure s (ι r, ι s) None. Together with (2) this implies completeness of eqv: Theorem 3 (in rexpdfa). L r = L s = eqv r s. This is the end of ll considertions out equivlence of regulr expressions. The rest of the pper merely needs to focus on vrious methods for turning regulr expressions into finite utomt in the sense of rexpdfa. Note tht M L defined ove constitutes first vlid interprettion of rexpdfa. The proof of fin requires the Myhill-Nerode theorem. interprettion M: rexpdfa where ι r = L r δ L = D L o L = [] L L L = L This interprettion is not executle ecuse neither its next-step function D (eing sed on infinite sets of words defined y set comprehension) nor the equlity on σ = αlng (which is needed for the closure computtion) is executle. (2)

5 4 Derivtives In 1964, Brzozowski [7] showed how to compute left quotients syntcticlly s derivtives of regulr expressions. Derivtives hve een rediscovered in proof ssistnts y Kruss nd Nipkow [19] nd Coqund nd Siles [10]. Our first executle instntitions of the frmework reuse infrstructure from erlier formliztions in Iselle [19, 26]. A refinement of Brzozowski s pproch, prtil derivtives, ws introduced y Antimirov [2] nd formlized y Moreir et l. [21] in Coq nd y Wu et l. [27] in Iselle. Prtil derivtives operte on finite sets of regulr expressions. They cn e viewed either s nondeterministic utomton with regulr expressions s sttes or s the corresponding deterministic utomton otined y the suset construction. In the following, we integrte the two notions in our frmework nd show how derivtives cn e used to simulte prtil derivtives without invoking sets explicitly. 4.1 Brzozowski s Derivtives Given letter c nd regulr expression r, the (Brzozowski) derivtive der :: α αrexp αrexp of r w.r.t. is defined y primitive recursion: der _ 0 = 0 der _ 1 = 0 der (A x) = if x = then 1 else 0 der (r + s) = der r + der s der (r s) = if nullle r then (der r s) + der s else der r s der (r ) = der r r It follows y induction on r tht the lnguge of the derivtive der r is exctly the left quotient D (L r). This property corresponds exctly to the ssumption δl of the locle rexpda. Hence it suggests the following interprettion: interprettion rexpda where ι r = r δ r = der r o r = nullle r L r = L r Unfortuntely, the sound equivlence checker tht is produced y this interprettion is useless in prctice, ecuse it will rrely terminte. For exmple, the utomton constructed from the regulr expression is infinite, s ll derivtives w.r.t. words n re distinct: fold der 1 = 1 ; fold der n+1 = 0 + fold der n. Fortuntely, Brzozowski showed tht there re finitely mny equivlence clsses of derivtives modulo ssocitivity, commuttivity nd idempotence (ACI) of the + constructor. We prove tht the numer of distinct derivtives of r modulo ACI is finite: finite {[fold der w r] w (Σ r) } where [r] = {s r s} denotes the equivlence clss of r nd the ACI equivlence is defined inductively s follows. r + (s + t) (r + s) + t r + s s + r r + r r r s r s s t r r s r r t r 1 s 1 r 2 s 2 r 1 s 1 r 2 s 2 r s r 1 + r 2 s 1 + s 2 r 1 r 2 s 1 s 2 r s ACI-equivlent regulr expressions r s hve the sme toms nd sme lnguges, nd their equivlence is preserved y the derivtive: der r der s for ll Σr.

6 This enles the following interprettion tht opertes on ACI equivlence clsses. We otin first totlly correct nd complete equivlence checker D.eqv in Iselle/HOL. interprettion D : rexpdfa where ι r = [r] δ [r] = [der r] o[r] = nullle r [(1 ) + 0] L[r] = L r [((0 + 1 ) + 0) + 0] [((0 + 0 ) + 1) + 0] [ ] [(0 ) + 1] [((0 + 0 ) + 0) + 0] Fig. 1: Derivtive utomton modulo ACI for Techniclly, the formliztion defines quotient type [18] of regulr expressions modulo ACI to represent equivlence clsses nd uses Lifting nd Trnsfer [17] to lift opertions on regulr expressions to opertions on equivlence clsses. The ove presenttion of definitions of the locle prmeters y pttern mtching on equivlence clsses resemles the code generted y Iselle for quotients ( pseudo-constructor [14], [_], wrps concrete representtive r), rther thn the ctul definitions y Lifting. Since the equivlence checker must compre equivlence clsses, the code genertion for quotients requires n executle equlity (i.e. decision procedure for - equivlence). We chieve this through n ACI normliztion function _ tht mps regulr expression r to cnonicl representtive of [r] y sorting ll summnds w.r.t. n ritrry fixed liner order while removing duplictes. The definition of _ employs smrt (simplifying) constructor, whose equtions re mtched sequentilly. 0 = 0 1 = 1 A = A r + s = r s r s = r s r = r (r + s) t = r (s t) r (s + t) = if r = s then s + t else if r s then r + (s + t) else s + (r t) r s = if r = s then r else if r s then r + s else s + r We otin n executle decision procedure for ACI equivlence: r s r = s. This mkes D.eqv executle, yielding verified code in different functionl progrmming lnguges vi Iselle s code genertor. Yet, the performnce of the generted code is disppointing. Fig. 1 shows why: Derivtions clutter concrete representtives with duplicted summnds. Further derivtion steps perform the sme computtion repetedly nd hence ecome incresingly expensive. This ottleneck is voided y tking cnonicl ACI-normlized representtives s sttes yielding second interprettion. interprettion D: rexpdfa where ι r = r δ r = der r o r = nullle r 0 + (1 ) L r = L r 1 + (0 ) 0 + (0 + 1 ) 0 + (1 + (0 ) ) 0 + (0 ) Fig. 2: ACI-normlized derivtive utomton for

7 A few points re worth mentioning here: First, D does not use the quotient type it opertes directly on cnonicl representtives nd therefore cn use structurl equlity for comprison (rther thn ). Second, the interprettions D nd D yield structurlly the sme utomt, lthough with different lels. Fig. 2 shows the utomton produced y D for. This oservtion which enles us to reuse the techniclly involved proof of D.fin to dischrge D.fin relies crucilly on our normliztion function _ eing idempotent nd well-ehved w.r.t. derivtives: Lemm 4. We hve r = r nd der r = der r for ll Σ r. The utomton from Fig. 2 shows tht the stte lels still contin superfluous informtion, notly in the form of 0s nd 1s. A corser reltion thn -equivlence, denoted, dresses this concern. We omit the strightforwrd inductive definition of, which cncels 0s nd 1s where possile nd tkes the ssocitivity of conctention into ccount. Corseness ([r] [r] ) together with D.fin implies finiteness of equivlence clsses of derivtives modulo : finite {[fold der w r] w (Σ r) }. As efore, to void working with equivlence clsses, we use recursively defined -normliztion function _ similr to _ (it corresponds to the norm function from the formliztion y Kruss nd Nipkow [19]). However, _ (lso ) is not well-ehved w.r.t. derivtives: for exmple, der (( + 1) ( )) der ((( + 1) ( )) ). The normliztion would need to tke the distriutivity of over + into ccount to prevent this disequlity, ut even with this ddition forml proof of well-ehvedness seems difficult. Furthermore, our evlution (Sect. 6) suggests tht not too much energy should e invested in finding this proof. Thus, the following interprettion gives only prtil correctness result. interprettion N: rexpda where ι r = r δ r = der r o r = nullle r L r = L r 1 0 Fig. 3: Normlized derivtive utomton for In prctice, we did not find n input for which N would construct n infinite utomton. For the exmple it even yields the miniml utomton shown in Fig Prtil Derivtives Prtil derivtives split certin +-constructors into sets of regulr expressions, thus cpturing ACI equivlence directly in the dt structure. The utomton construction for regulr expression r strts with the singleton set {r}. More precisely, prtil derivtives pder :: α α rexp (α rexp) set re defined recursively s follows: pder _ 0 = {} pder _ 1 = {} pder (A x) = if x = then {1} else {} pder (r + s) = pder r pder s pder (r s) = if nullle r then (pder r s) pder s else pder r s pder (r ) = pder r r

8 Aove, R s is used s shorthnd nottion for {r s r R}. The definition yields the chrcteristic property of prtil derivtives y induction on r: D (L r) = s pder r L s Following this chrcteristic property, we cn interpret the locle rexpdfa. The utomton constructed y P for our running exmple is shown in Fig. 4. interprettion P: rexpdfa where ι r = {r} δ R = r R pder r o R = r R. nullle r L R = r R L r { } {(1 ) } {1} {} Fig. 4: Prtil derivtive utomton for The ssumptions of rexpda (inherited y rexpdfa) re esy to dischrge. Just s for Brzozowski derivtives, only the proof of finiteness of the rechle stte spce P.fin poses chllenge. We were le to reuse the proof y Wu et l. [27] who show finiteness when proving one direction of the Myhill-Nerode theorem. Compred with the proof of D.fin, the forml resoning out prtil derivtives ppers to e more succinct. There is direct connection etween pder nd der tht seems not to hve een covered in the literture. It is est expressed in terms of recursive function pset :: αrexp (α rexp) set tht trnsltes derivtives to prtil derivtives: pset (der r) = pder r. pset 0 = {} pset (r + s) = pset r pset s pset 1 = {1} pset (r s) = pset r s pset (A x) = {A x} pset (r ) = {r } A finite set R of regulr expressions cn e represented uniquely y single regulr expression R, sum ordered w.r.t.. Hence, we hve pset (der r) = pder r, mening tht we cn devise normliztion function r = pset r tht llows us to simulte prtil derivtives while operting on plin regulr expressions. Alterntively, _ cn e defined using smrt constructors (with sequentilly mtched equtions): 0 = 0 1 = 1 A = A r + s = r s r s = r s r = r 0 r = 0 (r + s) t = (r s) (s t) r s = s t 0 r = r r 0 = r (r + s) t = r (s t) r (s + t) = if r = s then s + t else if r s then r + (s + t) else s + (r t) r s = if r = s then r else if r s then r + s else s + r This definition llows to contrst the implicit quotienting performed y prtil derivtives with the qoutienting modulo ACI equivlence ( ). They turn out to e incomprle: _ does not simplify the second rgument of conctention nd the rgument of itertion, ut erses 0s nd uses left distriutivity. Finlly, we otin lst derivtive-sed interprettion using the chrcteristic property der r = (pder r) nd P.fin to dischrge the finiteness ssumption fin.

9 interprettion PD: rexpdfa where ι r = r δ r = der r o r = nullle r L r = L r Whenever P yields n utomton for r with sttes leled with finite sets of regulr expressions X i, PD constructs structurlly the sme utomton for r leled with X i. 5 Mrked Regulr Expressions One of the oldest methods for converting regulr expression into n utomton is sed on the ide of identifying the sttes of the utomton with positions in the regulr expression. Both McNughton nd Ymd [20] nd Glushkov [13] mrk the toms in regulr expression with numers to identify positions uniquely. In this section, we formlize two recent reincrntions of this pproch due to Fischer et l. [11] nd Asperti [3]. They re sed on the reliztion tht in functionl progrmming setting, it is most convenient to represent positions in regulr expression y mrking some of its toms. First we define n infrstructure for working with mrked regulr expressions. Then we define nd relte oth reincrntions in terms of this infrstructure. Mrked regulr expressions re formlized y the following type synonym (where the vlue True denotes mrked tom) α mrexp = (ool α)rexp We convert esily etween rexp nd mrexp with the help of mp_rexp, the mp function on regulr expressions: strip = mp_rexp snd emtpy_ mrexp = mp_rexp (λr. (Flse, r)) The lnguge L m :: α mrexp αlng of mrked regulr expression is the set of words tht strt t some mrked tom: L m 0 = {} L m 1 = {} L m (A (m, )) = if m then {[]} else {} L m (r + s) = L m r L m s L m (r s) = (L m r L (strip s)) L m s L m (r ) = L m r L (strip r) The function finl :: α mrexp ool tests if some tom t the end of given regulr expression is mrked: finl 0 = Flse finl 1 = Flse finl (A (m, )) = m finl (r + s) = (finl r finl s) finl (r s) = (finl s nullle s finl r) finl (r ) = finl r

10 Mrks re moved round regulr expression y two opertions. The function red r unmrks ll toms in r except : red :: α α mrexp α mrexp red = mp_rexp (λ(m, x). (m = x, x)) Its chrcteristic lemm is tht it restricts L m r to words whose hed is : L m (red r) = {w L m r w [] hd w = } The function follow m r moves ll mrks in r to the next tom, much like n ε-closure; the mrk m is pushed in from the left: follow :: ool α mrexp α mrexp follow m 0 = 0 follow m 1 = 1 follow m (A (_, )) = A (m, ) follow m (r + s) = follow m r + follow m s follow m (r s) = follow m r follow (finl r m nullle r) s follow m (r ) = (follow (finl r m) r) The chrcteristic lemm out follow shows tht the mrks re moved forwrd, therey chopping off the first letter (in the generted lnguge), nd tht the prmeter m indictes whether every first tom should e mrked: L m (follow m r) = {tl w w L m r} (if m then L (strip r) else {}) {[]} 5.1 Mrk After Atom In the work of McNughton-Ymd-Glushkov, the mrk indictes which tom hs just een red, i.e. the mrk is locted fter the tom. Therefore the initil stte is specil ecuse nothing hs een red yet. Thus we express the sttes of the utomton s pir of oolen (True mens tht nothing hs een red yet) nd mrked regulr expression. The oolen cn e viewed s mrk in front of the utomton. (Alterntively, one could work with n explicit strt symol in front of the regulr expression.) We interpret the locle rexpdfa s follows: interprettion A: rexpdfa where ι r = (True, emtpy_ mrexp r) δ (m, r) = (Flse, red (follow m r)) o (m, r) = (finl r m nullle r) L (m, r) = L m (follow m r) (if o (m, r) then {[]} else {}) The definition of δ expresses tht we first uild the ε-closure strting from the mrked toms (vi follow) nd then red the next tom. With the chrcteristic lemms out red nd follow (nd few uxiliry lemms), the locle ssumptions re esily proved. This yields our first version of utomt sed on mrked regulr expressions. Finiteness of the rechle prt of the stte spce is proved vi the lemm fold δ w (ι r) {True, Flse} mrexps r

11 where mrexps :: αrexp (α mrexp)set mps regulr expression to the finite set of ll its mrked vrints, i.e. mrexps r = {r strip r = r}; its ctul recursive definition is strightforwrd nd omitted. Now we tke closer look t the work of Fischer et l. [11], which inspired the preceding formliztion. They present numer of (not formlly verified) mtching lgorithms on mrked regulr expressions in Hskell tht follow McNughton-Ymd- Glushkov. This is their sic trnsition function: shift :: ool α mrexp α α mrexp shift _ 0 _ = 0 shift _ 1 _ = 1 shift m (A (_, x)) c = A (m (x = c), x) shift m (r + s) c = shift m r c + shift m s c shift m (r s) c = shift m r c shift (finl r m nullle r) s c shift m (r ) c = (shift (finl r m) r c) A simple induction proves tht their shift is our δ: shift m r x = red x (follow m r) Thus we hve verified their shift function. Fischer et l. optimize shift further, which is still qudrtic due to the clls of the recursive functions finl nd nullle. They simply cche the vlues of finl nd nullle t ll nodes of regulr expression y dding dditionl fields to ech constructor. We hve verified this optimiztion step s well, yielding nother interprettion A 2 (omitted here). 5.2 Mrk Before Atom Insted of imgining the mrk to e fter n tom, it cn lso e viewed to e in front of it, i.e. it mrks possile next toms. This is somewht dul to the McNughton-Ymd- Glushkov construction. It leds to the following interprettion of the rexpda locle: interprettion B: rexpdfa where ι r = (follow True (emtpy_ mrexp r), nullle r) δ (r, m) = let r = red r in (follow Flse r, finl r ) o (r, m) = m L (r, m) = L m r (if m then {[]} else {}) The definition of δ expresses tht we first red n tom nd then uild the ε-closure. The ssumptions of rexpda nd rexpdfa re proved esily just like in the previous interprettion with mrked regulr expressions. The interesting point is tht this hppens to e the lgorithm formlized y Asperti [3]. Although he sys tht he hs formlized McNughton-Ymd, he ctully formlized the dul lgorithm. This is not esy to see ecuse Asperti s formliztion is considerly more involved thn ours, with mny uxiliry functions. Strictly speking, his lgorithm is vrition of ours tht produces the sme utomt. The complete proof of this fct cn e found elsewhere [16]. Becuse of the size of Asperti s formliztion, there is not enough spce here to give the detiled equivlence proof. However, we cn tke step towrds his formultion nd merge follow nd red into one function move :: α α mrexp ool α mrexp, the nlogue of his homonymous function:

12 ( ) ( ) ( ) ( ) ( ) ( ) Fig. 5: Mrked regulr expression utomt (A left, B right) for move _ 0 _ = 0 move _ 1 _ = 1 move c (A (_, x)) m = A (m, x) move c (r + s) m = move c r m + move c s m move c (r s) m = move c r m move c s (finl1 r c m nullle r) move c (r ) m = (move c r (finl1 r c m)) where finl1 is n uxiliry recursive function (not shown here) with the chrcteristic property tht finl1 r c = finl (red c r). A simple induction proves tht move comines follow nd red s in δ: move c r m = follow m (red c r) The function move hs qudrtic complexity for the sme reson s shift. Unfortuntely, it cnnot e mde liner with the sme ese s for shift. The prolem is tht we need to cche the vlue of finl1 r c in the previous step, efore we know c. We solve this y cching the set of ll letters c tht mke finl1 r c true. In the worst cse, the whole lphet must e stored in certin inner nodes. However, for n lphet of fixed size this gurntees liner time complexity. This optimiztion constitutes lst interprettion B 2. Even for fixed lphet, Asperti s move hs qudrtic complexity when fced with tower of strs: ech recursive cll of move cn trigger cll of function eclose, which hs liner complexity. Asperti imed for compct proofs, not mximl efficiency. 5.3 Comprison The two constructions my look similr, ut they do not produce isomorphic utomt. Considering our running exmple, we disply the mrk y efore or fter the tom. The two resulting utomt re shown in Fig. 5. There re specil sttes tht cnnot e denoted y mrking toms only: r in A s utomton is the completely unmrked regulr expression tht is the initil stte nd r in B s utomton is finl stte. It turns out tht the efore utomton is homomorphic imge of the fter utomton. To verify this we specify the homomorphism ϕ(m, r) = (follow m r, A.o (m, r)) nd prove tht it preserves initil sttes nd commutes with the trnsition function: ϕ(a.ι r) = B.ι r ϕ(a.δ s) = B.δ (ϕ s) ϕ(fold A.δ w s) = fold B.δ w (ϕ s) A direct consequence is tht Asperti s efore construction lwys genertes utomt with t most s mny sttes s the McNughton-Ymd-Glushkov construction. Formlly, in the context of locle rexpda we hve defined n executle computtion of the rechle stte spce {fold δ w (ι r) w (set s) } of the utomton: rechle s r = snd (the (rtc (λ_. True) (λ s. mp (λ. δ s) s) (ι r)))

13 where r is the initil regulr expression, s is the lphet, nd the (Some x) = x. Theorem 5. B.rechle s r A.rechle s r where _ is the crdinlity of set. In erly drfts of this pper, we only conjectured the ove sttement nd unsuccessfully tried to refute it with Iselle s Quickcheck fcility [8]. Lter, Helmut Seidl hs communicted n informl proof using the ove homomorphism to us. Let us revite the sttement of Thm. 5 to n n. One my think tht n is only slightly lrger thn n, ut it seems tht n nd n re more thn constnt summnd prt: for two-element lphet Quickcheck could refute n n +k even for k = Empiricl Comprison We compre the efficiency w.r.t. oth mtching nd deciding equivlence of the Stndrd ML code generted from eight descried interprettions: -normlized derivtives (D), -normlized derivtives (N), prtil derivtives (P), derivtives simulting prtil derivtives (PD), mrk fter tom (A), mrk fter tom with cching (A 2 ), mrk efore tom (B), nd mrk efore tom with cching (B 2 ). The interprettion using the quotient type for derivtives (D ) is not in this list, s it is clerly superseded y D. The results of the evlution, performed on n Intel Core i7-2760qm mchine with 8GB of RAM, re shown in Fig. 6. Solid lines depict the four derivtive-sed lgorithms. Dshed lines re used for the lgorithms sed on mrked regulr expressions. The first two tests, MATCH-R nd MATCH-L, mesure the time required to mtch the word n ginst the regulr expression ( + 1) n n stndrd enchmrk lso used y Fischer et l. [11]. The difference etween the two tests is the definition of r n. MATCH-R defines it s the n-fold conctention ssocited to the right: r 4 = r (r (r r)), wheres MATCH-L ssocites to the left: r 4 = ((r r) r) r. In oth tests, mrked regulr expressions outperform derivtives y fr. The normliztion performed y the derivtive-sed pproches (required to otin finite numer of sttes for the equivlence check) decelertes the computtion of the next stte. Mrked regulr expressions enefit from fst next stte computtion. The test MATCH-L exhiits the qudrtic nture of the unoptimized mtchers A nd B (their curves re lmost identicl nd therefore hrd to distinguish in Fig. 6). In contrst, A 2 nd B 2 perform eqully well in oth tests, A 2 eing pproximtely 1.5 times fster due to lighter cche nnottions. The next test goes ck to Antimirov [1]: We mesure the time (with timeout of ten seconds) for proving the equivlence of nd ( n 1 ) ( n ). Agin two tests, EQ-R nd EQ-L, distinguish the ssocitivity of conctention in r n. Here, the derivtive-sed equivlence checkers (except for D) perform etter then the ones sed on mrked regulr expressions. In prticulr, oth version of prtil derivtives, P nd PD, outperform N since this exmple ws crfted y Antimirov to demonstrte the strength of prtil derivtives, this is not wholly unexpected. Compring EQ-R nd EQ-L, the ssocitivity rely influences the runtime. Finlly, to void is towrds prticulr lgorithm, we hve devised the rndomized test EQ-RND. There we mesure the verge time (with timeout of ten seconds) to prove the equivlence of r with itself for 100 rndomly generted expressions with n inner nodes (+,, or ). Proving r r is of course trivil tsk, ut our lgorithms do not

14 4 (MATCH-R) 4 (MATCH-L) Time (s) 2 Time (s) n n (EQ-R) (EQ-L) 8 8 Time (s) n Time (s) n Time (s) 4 2 (EQ-RND) n D N P PD B B 2 A A 2 Fig. 6: Evlution results stop the explortion when the stte of the product utomton is pir of two equl sttes. This optimiztion, which is must for ny prcticl lgorithm, is the first step towrds the rewrding usge of isimultion up to equivlence (or even up to congruence) [5]. Without ny such optimiztion, the tsk of proving r r mounts to enumerting ll derivtives of r, which is exctly wht we wnt to compre. To generte rndom regulr expression we use the infrstructure of SpecCheck [25] Quickcheck clone for Iselle/ML. For computing the verge, timeout counts s 10 second (lthough the ctul computtion would likely hve tken longer) n pproximtion tht skews the curves to converge to the mrgin of 10 seconds. We stopped mesuring method for incresing n when the verge pproched 5 seconds. The results of EQ-RND re summrized s follows: D N P,PD A,A 2,B,B 2, where X Y mens tht Y is n order of mgnitude fster thn X. The lgorithm P is noticely slower thn PD voiding sets reduces the overhed. Among A, A 2, B, B 2, Asperti s unoptimized lgorithm B performs est y nrrow mrgin. Regulr expressions where the cching overhed pys off re rre nd therefore not visile in the rndomized test results. The sme holds for expressions where B produces much smller utomt thn A (e.g. the counterexmple to n n from Susect. 5.3). Our evlution shows tht A 2 is the est choice for mtching. For equivlence checking, the winner is not s cler cut: B (especilly when pplied to normlized input to void qudrtic runtime without cching) nd PD seem to e the est choices.

15 7 Extensions Brzozowski s derivtives re esily extendle to regulr expressions intersection nd negtion indeed Brzozowski performed the extension right from the strt [7]. The numer of such extended derivtives is still finite when quotiented modulo ACI. We [26] hve recently further extended derivtives to regulr expressions extended with projection, otining verified decision procedures for the equivlence of those extended regulr expressions nd for mondic second-order logics over finite words. The closure computtion nd its correctness proof follow Kruss nd Nipkow [19]. Extending prtil derivtives with intersection nd negtion is more involved [9]. An dditionl lyer of sets must e used for intersections, i.e. the sttes of our utomton would then e sets of sets of regulr expressions. In Sect. 6, we hve seen tht lredy one lyer of sets incurs some overhed. Hence, the view on prtil derivtives s derivtives followed y some normliztion is expected to e even more profitle for the extension. The extension of prtil derivtives with projection is n esy exercise. It is uncler how to extend mrked regulr expressions to hndle negtion nd intersection. The numer of possile mrkings for regulr expression of lphetic width n is 2 n. However, there exist regulr expressions of lphetic width n using intersection, whose miniml utomt hve 2 2n sttes [12]. 8 Conclusion We hve shown tht ll the previously pulished verified decision procedures for equivlence of regulr expressions tht operte on regulr expressions directly cn ll e expressed s instnces of generic utomton-inspired frmework. The correctness proofs decompose into generic prt tht is proved once nd for ll in the frmework nd few specific properties tht need to e proved for ech instnce. The frmework cters for meningful comprison of the performnce of the vrious instnces. Mrked regulr expressions re superior on verge ut prtil derivtives cn outperform them in specific cses. The Iselle theories re ville online [23]. Acknowledgment. We thnk Andre Asperti nd Sestin Fischer for commenting on fine points of their work nd Helmut Seidl for contriuting n informl proof of Thm. 5. Jsmin Blnchette, Andrei Popescu nd three nonymous reviewers helped to improve the presenttion through numerous suggestions. The second uthor is supported y the doctorte progrm 1480 (PUMA) of the Deutsche Forschungsgemeinschft (DFG). References 1. Antimirov, V.: Prtil derivtives of regulr expressions nd finite utomt constructions. In: Myr, E.W., Puech, C. (eds.) STACS 95. LNCS, vol. 900, pp Springer (1995) 2. Antimirov, V.: Prtil derivtives of regulr expressions nd finite utomton constructions. Theor. Comput. Sci. 155(2), (1996) 3. Asperti, A.: A compct proof of decidility for regulr expression equivlence. In: Beringer, L., Felty, A. (eds.) ITP LNCS, vol. 7406, pp Springer (2012) 4. Bllrin, C.: Interprettion of locles in Iselle: Theories nd proof contexts. In: Borwein, J.M., Frmer, W.M. (eds.) MKM LNCS, vol. 4108, pp Springer (2006)

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